High Speed and Low-Complexity Hardware Architectures for Elliptic Curve-Based Crypto-Processors

The elliptic curve cryptography (ECC) has been identified as an efficient scheme for public-key cryptography. This thesis studies efficient implementation of ECC crypto-processors on hardware platforms in a bottom-up approach. We first study efficient and low-complexity architectures for finite field multiplications over Gaussian normal basis (GNB). We propose three new low-complexity digit-level architectures for finite field multiplication. Architectures are modified in order to make them more suitable for hardware implementations specially focusing on reducing the area usage. Then, for the first time, we propose a hybrid digit-level multiplier architecture which performs two multiplications together (double-multiplication) with the same number of clock cycles required as the one for one multiplication. We propose a new hardware architecture for point multiplication on newly introduced binary Edwards and generalized Hessian curves. We investigate higher level parallelization and lower level scheduling for point multiplication on these curves. Also, we propose a highly parallel architecture for point multiplication on Koblitz curves by modifying the addition formulation. Several FPGA implementations exploiting these modifications are presented in this thesis. We employed the proposed hybrid multiplier architecture to reduce the latency of point multiplication in ECC crypto-processors as well as the double-exponentiation. This scheme is the first known method to increase the speed of point multiplication whenever parallelization fails due to the data dependencies amongst lower level arithmetic computations. Our comparison results show that our proposed multiplier architectures outperform the counterparts available in the literature. Furthermore, fast computation of point multiplication on different binary elliptic curves is achieved

Multi-Base Chains for Faster Elliptic Curve Cryptography

This research addresses a multi-base number system (MBNS) for faster elliptic curve cryptography (ECC). The emphasis is on speeding up the main operation of ECC: scalar multiplication (tP). Mainly, it addresses the two issues of using the MBNS with ECC: deriving optimized formulas and choosing fast methods. To address the first issue, this research studies the optimized formulas (e.g., 3P, 5P) in different elliptic curve coordinate systems over prime and binary fields. For elliptic curves over prime fields, affine Weierstrass, Jacobian Weierstrass, and standard twisted Edwards coordinate systems are reviewed. For binary elliptic curves, affine, Lambda-projective, and twisted mu4-normal coordinate systems are reviewed. Additionally, whenever possible, this research derives several optimized formulas for these coordinate systems. To address the second issue, this research theoretically and experimentally studies the MBNS methods with respect to the average chain length, the average chain cost, and the average conversion cost. The reviewed MBNS methods are greedy, ternary/binary, multi-base NAF, tree-based, and rDAG-based. The emphasis is on these methods\u27 techniques to convert integer t to multi-base chains. Additionally, this research develops bucket methods that advance the MBNS methods. The experimental results show that the MBNS methods with the optimized formulas, in general, have good improvements on the performance of scalar multiplication, compared to the single-base number system methods

Arithmetic of τ\tau-adic Expansions for Lightweight Koblitz Curve Cryptography

Koblitz curves allow very efficient elliptic curve cryptography. The reason is that one can trade expensive point doublings to cheap Frobenius endomorphisms by representing the scalar as a tau-adic expansion. Typically elliptic curve cryptosystems, such as ECDSA, also require the scalar as an integer. This results in a need for conversions between integers and the tau-adic domain, which are costly and hinder the use of Koblitz curves on very constrained devices, such as RFID tags, wireless sensors, or certain applications of the Internet of things. We provide solutions to this problem by showing how complete cryptographic processes, such as ECDSA signing, can be completed in the tau-adic domain with very few resources. This allows outsourcing conversions to a more powerful party. We provide several algorithms for performing arithmetic operations in the tau-adic domain. In particular, we introduce a new representation allowing more efficient and secure computations compared to the algorithms available in the preliminary version of this work from CARDIS 2014. We also provide datapath extensions with different speed and side-channel resistance properties that require areas from less than one hundred to a few hundred gate equivalents on 0.13-mu m CMOS. These extensions are applicable for all Koblitz curves.Peer reviewe

Reconfigurable elliptic curve cryptography

Elliptic Curve Cryptosystems (ECC) have been proposed as an alternative to other established public key cryptosystems such as RSA (Rivest Shamir Adleman). ECC provide more security per bit than other known public key schemes based on the discrete logarithm problem. Smaller key sizes result in faster computations, lower power consumption and memory and bandwidth savings, thus making ECC a fast, flexible and cost-effective solution for providing security in constrained environments. Implementing ECC on reconfigurable platform combines the speed, security and concurrency of hardware along with the flexibility of the software approach. This work proposes a generic architecture for elliptic curve cryptosystem on a Field Programmable Gate Array (FPGA) that performs an elliptic curve scalar multiplication in 1.16milliseconds for GF (2163), which is considerably faster than most other documented implementations. One of the benefits of the proposed processor architecture is that it is easily reprogrammable to use different algorithms and is adaptable to any field order. Also through reconfiguration the arithmetic unit can be optimized for different area/speed requirements. The mathematics involved uses binary extension field of the form GF (2n) as the underlying field and polynomial basis for the representation of the elements in the field. A significant gain in performance is obtained by using projective coordinates for the points on the curve during the computation process